Advanced Courses in Mathematics - CRM Barcelona

Variable Lebesgue Spaces and Hyperbolic Systems

Authors: Cruz-Uribe, D., Fiorenza, A., Ruzhansky, M.V., Wirth, J.

Editors: Tikhonov, Sergey (Ed.)

  • Features a concise introduction to variable Lebesgue spaces requiring only basic knowledge of analysis
  • Includes an easy-to-read introduction to the classical problems as well as to recent developments in the asymptotic theory for hyperbolic equations
  • The presentation of the material starts at a basic level but gives several deeper insights into different aspects of the theories up to the most recent developments
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eBook $24.99
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  • ISBN 978-3-0348-0840-8
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  • Immediate eBook download after purchase
Softcover $34.99
price for USA
  • ISBN 978-3-0348-0839-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
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About this Textbook

This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts.

Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some more technical proofs and background material omitted.

Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations and systems with time-dependent coefficients. First, an overview of known results is given for general scalar hyperbolic equations of higher order with constant coefficients. Then strongly hyperbolic systems with time-dependent coefficients are considered. A feature of the described approach is that oscillations in coefficients are allowed. Propagators for the Cauchy problems are constructed as oscillatory integrals by working in appropriate time-frequency symbol classes. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.

Table of contents (11 chapters)

  • Introduction and Motivation

    Cruz-Uribe, David (et al.)

    Pages 3-9

  • Properties of Variable Lebesgue Spaces

    Cruz-Uribe, David (et al.)

    Pages 11-34

  • The Hardy–Littlewood Maximal Operator

    Cruz-Uribe, David (et al.)

    Pages 35-56

  • Extrapolation in Variable Lebesgue Spaces

    Cruz-Uribe, David (et al.)

    Pages 57-82

  • Introduction

    Cruz-Uribe, David (et al.)

    Pages 93-98

Buy this book

eBook $24.99
price for USA (gross)
  • ISBN 978-3-0348-0840-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $34.99
price for USA
  • ISBN 978-3-0348-0839-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Variable Lebesgue Spaces and Hyperbolic Systems
Authors
Editors
  • Sergey Tikhonov
Series Title
Advanced Courses in Mathematics - CRM Barcelona
Copyright
2014
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel
eBook ISBN
978-3-0348-0840-8
DOI
10.1007/978-3-0348-0840-8
Softcover ISBN
978-3-0348-0839-2
Series ISSN
2297-0304
Edition Number
1
Number of Pages
IX, 170
Number of Illustrations and Tables
5 b/w illustrations
Topics